WASHINGTON — Now that the U.S. House and Senate have passed budgets that are far apart, the focus shifts to two lawmakers who don’t share many ideas but did share a recent breakfast.

After getting word in December that she’d be heading the Senate Budget Committee in 2013, Democrat Patty Murray picked up the check when she had breakfast with Republican Rep. Paul Ryan in the Senate Dining Room.

They’re nearly a generation apart with little in common, but the Washington Democrat wanted to meet privately to size up the man who would soon become her latest adversary on Capitol Hill.

The 62-year-old Murray, who wants to raise taxes on the wealthy and who always ranks as one of the top Senate liberals, had never met the 43-year-old Ryan, the up-and-coming chairman of the House Budget Committee from Wisconsin, who shot to the top of the Republican ranks and the No. 2 spot on the 2012 GOP presidential ticket with his aggressive calls for spending cuts and no new taxes.

“I thought it was important to find out who he was and get to know him personally,” Murray recalled. “At the end of the day, we’re going to have to come together and figure out a path forward.”

Murray led the so-called congressional supercommittee that failed to produce a deficit-reduction plan in 2011, setting the stage for the automatic spending cuts that took effect March 1. But she’s upbeat that her new starring role in the budget quagmire will produce a different result, even ending the across-the-board cuts caused by sequestration before they hit too hard.

Much will depend on her budding relationship with Ryan, even though the two are busy trading barbs, at least for now. Just Thursday, she labeled his budget a set of “failed priorities.” He said hers “(clings) to the status quo.”

Ultimately, Murray said, one of her biggest tasks will be to ease the partisan rhetoric that has marked the budget fights, “the challenge of calming that down,” as she put it.

Negotiations between the two could begin in earnest in April, as soon as Congress returns from its traditional Easter break. The Republican-controlled House of Representatives passed Ryan’s budget plan Thursday without a single Democrat in support. The Democratic-run Senate signed off on Murray’s plan Saturday by a vote of 50-49.

Murray said she and Ryan have exchanged phone calls since their breakfast meeting and have come to appreciate each other’s roles. “We have a relationship that understands that we both have an important job on our side to get budgets out — and all the challenges that come with that,” she said.

In a briefing with reporters earlier this month, Ryan said he thinks the two sides have an incentive to work together.

“We’re going to have to talk to each other to get an agreement about how to delay a debt crisis,” he said.

And at a Capitol Hill news conference last week, he added: “Look, I’m very pleased that Patty Murray is attempting to pass a budget because we haven’t seen that attempt in a long time. … If she can pass a budget, then we actually have a process out in the public for the nation to see that gets us going down the path of solving problems.”

Murray said her new assignment is a big switch from leading the Senate Veterans Affairs Committee, which she headed in 2011 and 2012. There, she said, Republicans and Democrats were eager to work together to advance the popular causes of veterans.

To reach accord on a budget for 2014 and set the framework for the next 10 years, Murray said she’ll rely on the same skills she used in teaching 4-year-olds at the Shoreline Community College Co-op Preschool Program in her home state in the mid-1980s.

“It’s making sure that everybody’s voice is heard but also knowing when it’s time to say, ‘It’s my turn to talk,’” said Murray, a fourth-term senator first elected in 1992.

Hoping to save 750,000 jobs that Democrats say are threatened by the latest across-the-board spending cuts, Murray would replace them with a new plan to again raise taxes on millionaires while imposing a new set of cuts, including $240 billion from defense.

Her plan would create a new, $100 billion “economic recovery protection plan” to spend more to replace bridges, repair schools and bring broadband technology to more communities.

Overall, the Murray plan would cut deficits by another $1.85 trillion over the next decade, with $975 billion in new spending cuts and the same amount in new tax revenue.

Murray’s worried about the spending cuts in the sequester, which could cost 41,700 jobs and deliver a $3.4 billion hit to the Washington state economy, according to the state’s Office of Financial Management. On Monday, 250 pink slips went to workers at the Hanford nuclear waste site and more than 2,500 employees were warned of upcoming furloughs. Murray fears the loss of more money for schools, declining income for thousands of civilian employees at Joint Base Lewis-McChord, and delayed treatment for soldiers. And she said a loss of housing vouchers in King County could mean more homeless on the streets.

“It’s really scary,” Murray said.

Ryan’s plan, similar to what he campaigned on as Republican Mitt Romney’s vice presidential running mate last year, would cut deficits by $4.6 trillion over the next 10 years and reduce the annual growth in federal spending from 5 percent to 3.4 percent.

Ryan also wants to junk the health care plan passed by Congress in 2010 and allow seniors to choose whether to use Medicare or private insurance, beginning in 2024. He would then use the $716 billion cut from Medicare to help trim the federal deficit over the next decade.

Murray said she considers a balanced budget to be about balanced priorities, including spending on people. She said Ryan’s plan is filled with “gimmicks” that includes big tax breaks for the wealthy — a drop in the top tax rate from 39.6 percent to 25 percent — with no way to pay for them. She said the Republican plan would never balance but would hurt the middle class and destroy the federal safety net, despite Ryan’s assurances to the contrary.

“Look, anybody can write numbers on a piece of paper,” Murray said. “But the reality is somebody’s gonna pay more.”

Despite the gulf, Murray said that reaching a compromise with Ryan in a House-Senate conference committee “is definitely a possibility.” And she said that doing it in an orderly way would help calm the fears that Congress can do something other than respond to fiscal crises.

“Look, we’ve been debating this for months and years,” Murray said. “And one of the things I hear the most and I feel strongest about is that it’s time for our country to quit managing by crisis.”

It would mark a major change of pace for the Senate, which hasn’t passed a budget since 2009. At the same time, Congress has allowed the federal debt to explode to a record high of nearly $17 trillion.

Murray joked about the long delay when the Budget Committee voted 12-10 to approve her plan March 13, on the same day that the Catholic Church announced the selection of Pope Francis.

“I understand we have a new pope and a committee hearing to mark up a budget. History twice — so that’s good,” Murray said.

Murray faces many skeptics.

Senate Minority Leader Mitch McConnell of Kentucky said she had produced one of the “most left-wing budgets of the modern era.”

And Tim Phillips, the president of a government cost-cutting group, Americans For Prosperity, said Murray had created a budget “so outlandishly liberal it’s difficult to see where common ground can be found” with Republicans.

“She is a very traditional big-government liberal: The recipe for everything that ails us, in her view, is more government spending and programs and higher taxes to pay for some of it and then deficits to cover everything else,” he said.

Murray has backers, too, particularly in Washington state.

Among them: The Washington Wildlife and Recreation Coalition, whose executive director, Joanna Grist, cheered when Murray included $900 million in her budget for the federal government’s Land and Water Conservation Fund. That’s full funding for the program, and Grist said that would be only the second time that’s happened in 40 years. It could mean more money to protect waterfowl habitat at Nisqually National Wildlife Refuge, better recreational access to Mount Rainier National Park, improvements for Point Defiance Park in Tacoma and repairs for the community pool in Chehalis.

For Murray, the federal debt is no excuse to stop spending. It’s a philosophy shaped by her experience, growing up in a family of seven children that used food stamps after her father was diagnosed with multiple sclerosis and lost his job. She said her mother got help from the government to get an accounting degree so she could get a job, and she and six siblings got federal college loans.

“We were just an average middle-class family that got struck by something we didn’t see coming, a family that all of a sudden found itself in a very bad place,” Murray said. “We had a government that was there for us.”